I am taking a graduate course in probability and I am baffled by the definition of conditional expectation. We defined conditional expectation and are now proving many theorems that say that it works the same as normal expectation. This seems like a weird way of doing things.
I have read several related questions on this site but it did not answer my question. Specifically, I understand that conditional expectation has to be a random variable and I understand how a sigma-algebra conveys information.
Instead, I would first define conditional probability (probably in a way similar to the definition of conditional expectation). Then, I would take expectation with respect to this conditional distribution. That would have the advantage that I would not have to prove everything twice.
Why not do it like this?